Preconditioning the augmented Lagrangian method for instationary mean field games with diffusion

Roman Andreev (1, 2)

1 MOKAPLAN - Méthodes numériques pour le problème de Monge-Kantorovich et Applications en sciences sociales
2 LJLL - Laboratoire Jacques-Louis Lions

We discuss the application of the augmented Lagrangian method to the convex optimization problem of instationary variational mean field games with diffusion. The problem is first discretized with space-time tensor product piecewise polynomial bases. This leads to a sequence of linear problems posed on the space-time cylinder that are second order in the temporal variable and fourth order in the spatial variable. To solve these large linear problems with the preconditioned conjugate gradients method we propose a preconditioner that is based on a temporal transformation coupled with a spatial multigrid. This preconditioner is thus based on standard components and is particularly suitable for parallel computation. It is conditionally parameter-robust in the sense that the condition number of the preconditioned system is low for sufficiently fine temporal discretizations. Numerical examples illustrate the method.

Mots clés: multigrid; b-splines; preconditioning; space-time; discretization; mean field games; augmented Lagrangian; ALG2; ADMM

Domaines: Modélisation et simulation; Analyse numérique [math.NA]; Logiciel mathématique [cs.MS]; Analyse numérique [cs.NA]; Calcul parallèle, distribué et partagé [cs.DC]

Roman Andreev. Preconditioning the augmented Lagrangian method for instationary mean field games with diffusion. 2017. ⟨hal-01301282v2⟩